Dagger linear logic for categorical quantum mechanics
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Published:2021-11-16
Issue:
Volume:Volume 17, Issue 4
Page:
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ISSN:1860-5974
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Container-title:Logical Methods in Computer Science
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language:en
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Short-container-title:
Author:
Cockett Robin,Comfort Cole,Srinivasan Priyaa
Abstract
Categorical quantum mechanics exploits the dagger compact closed structure of
finite dimensional Hilbert spaces, and uses the graphical calculus of string
diagrams to facilitate reasoning about finite dimensional processes. A
significant portion of quantum physics, however, involves reasoning about
infinite dimensional processes, and it is well-known that the category of all
Hilbert spaces is not compact closed. Thus, a limitation of using dagger
compact closed categories is that one cannot directly accommodate reasoning
about infinite dimensional processes.
A natural categorical generalization of compact closed categories, in which
infinite dimensional spaces can be modelled, is *-autonomous categories and,
more generally, linearly distributive categories. This article starts the
development of this direction of generalizing categorical quantum mechanics. An
important first step is to establish the behaviour of the dagger in these more
general settings. Thus, these notes simultaneously develop the categorical
semantics of multiplicative dagger linear logic.
The notes end with the definition of a mixed unitary category. It is this
structure which is subsequently used to extend the key features of categorical
quantum mechanics.
Publisher
Centre pour la Communication Scientifique Directe (CCSD)
Subject
General Computer Science,Theoretical Computer Science